Euclidean group

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In mathematics, the Euclidean group E(n), also known as ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometries associated with the Euclidean metric, are called Euclidean motions.

These groups are among the oldest and most studied, at least in the cases of dimension 2 and 3 – implicitly, long before the concept of group was invented.

Overview

Dimensionality

The number of degrees of freedom for E(n) is n(n + 1)/2, which gives 3 in case n = 2, and 6 for n = 3. Of these, n can be attributed to available translational symmetry, and the remaining n(n − 1)/2 to rotational symmetry.

Direct and indirect isometries

There is a subgroup E+(n) of the direct isometries, i.e., isometries preserving orientation, also called rigid motions; they are the rigid body moves. These include the translations, and the rotations, which together generate E+(n). E+(n) is also called a special Euclidean group, and denoted SE(n).

The others are the indirect isometries. The subgroup E+(n) is of index 2. In other words, the indirect isometries form a single coset of E+(n). Given any indirect isometry, for example a given reflection R that reverses orientation, all indirect isometries are given as DR, where D is a direct isometry.

The Euclidean group for SE(3) is used for the kinematics of a rigid body, in classical mechanics. A rigid body motion is in effect the same as a curve in the Euclidean group. Starting with a body B oriented in a certain way at time t = 0, its orientation at any other time is related to the starting orientation by a Euclidean motion, say f(t). Setting t = 0, we have f(0) = I, the identity transformation. This means that the curve will always lie inside E+(3), in fact: starting at the identity transformation I, such a continuous curve can certainly never reach anything other than a direct isometry. This is for simple topological reasons: the determinant of the transformation cannot jump from +1 to −1.

The Euclidean groups are not only topological groups, they are Lie groups, so that calculus notions can be adapted immediately to this setting.

Relation to the affine group

The Euclidean group E(n) is a subgroup of the affine group for n dimensions, and in such a way as to respect the semidirect product structure of both groups. This gives, a fortiori, two ways of writing down elements in an explicit notation. These are:

1. by a pair (A, b), with A an n × n orthogonal matrix, and b a real column vector of size n; or
2. by a single square matrix of size n + 1, as explained for the affine group.

Details for the first representation are given in the next section.

In the terms of Felix Klein's Erlangen programme, we read off from this that Euclidean geometry, the geometry of the Euclidean group of symmetries, is therefore a specialisation of affine geometry. All affine theorems apply. The extra factor in Euclidean geometry is the notion of distance, from which angle can then be deduced.

Detailed discussion

Subgroup structure, matrix and vector representation

The Euclidean group is a subgroup of the group of affine transformations.

It has as subgroups the translational group T(n), and the orthogonal group O(n). Any element of E(n) is a translation followed by an orthogonal transformation (the linear part of the isometry), in a unique way:

${\displaystyle x\mapsto A(x+b)}$

where A is an orthogonal matrix

or an orthogonal transformation followed by a translation:

${\displaystyle x\mapsto Ax+b}$.

T(n) is a normal subgroup of E(n): for any translation t and any isometry u, we have

u−1tu

again a translation (one can say, through a displacement that is u acting on the displacement of t; a translation does not affect a displacement, so equivalently, the displacement is the result of the linear part of the isometry acting on t).

Together, these facts imply that E(n) is the semidirect product of O(n) extended by T(n). In other words, O(n) is (in the natural way) also the quotient group of E(n) by T(n):

O(n) ≅ E(n) / T(n).

Now SO(n), the special orthogonal group, is a subgroup of O(n), of index two. Therefore E(n) has a subgroup E+(n), also of index two, consisting of direct isometries. In these cases the determinant of A is 1.

They are represented as a translation followed by a rotation, rather than a translation followed by some kind of reflection (in dimensions 2 and 3, these are the familiar reflections in a mirror line or plane, which may be taken to include the origin, or in 3D, a rotoreflection).

We have:

SO(n) ≅ E+(n) / T(n).

Subgroups

Types of subgroups of E(n):

• Finite groups. They always have a fixed point. In 3D, for every point there are for every orientation two which are maximal (with respect to inclusion) among the finite groups: Oh and Ih. The groups Ih are even maximal among the groups including the next category.
• Countably infinite groups without arbitrarily small translations, rotations, or combinations, i.e., for every point the set of images under the isometries is topologically discrete. E.g. for 1 ≤ mn a group generated by m translations in independent directions, and possibly a finite point group. This includes lattices. Examples more general than those are the discrete space groups.
• Countably infinite groups with arbitrarily small translations, rotations, or combinations. In this case there are points for which the set of images under the isometries is not closed. Examples of such groups are, in 1D, the group generated by a translation of 1 and one of √2, and, in 2D, the group generated by a rotation about the origin by 1 radian.
• Non-countable groups, where there are points for which the set of images under the isometries is not closed. E.g. in 2D all translations in one direction, and all translations by rational distances in another direction.
• Non-countable groups, where for all points the set of images under the isometries is closed. E.g.
• all direct isometries that keep the origin fixed, or more generally, some point (in 3D called the rotation group)
• all isometries that keep the origin fixed, or more generally, some point (the orthogonal group)
• all direct isometries E+(n)
• the whole Euclidean group E(n)
• one of these groups in an m-dimensional subspace combined with a discrete group of isometries in the orthogonal (nm)-dimensional space
• one of these groups in an m-dimensional subspace combined with another one in the orthogonal (nm)-dimensional space

Examples in 3D of combinations:

• all rotations about one fixed axis
• ditto combined with reflection in planes through the axis and/or a plane perpendicular to the axis
• ditto combined with discrete translation along the axis or with all isometries along the axis
• a discrete point group, frieze group, or wallpaper group in a plane, combined with any symmetry group in the perpendicular direction
• all isometries which are a combination of a rotation about some axis and a proportional translation along the axis; in general this is combined with k-fold rotational isometries about the same axis (k ≥ 1); the set of images of a point under the isometries is a k-fold helix; in addition there may be a 2-fold rotation about a perpendicularly intersecting axis, and hence a k-fold helix of such axes.
• for any point group: the group of all isometries which are a combination of an isometry in the point group and a translation; for example, in the case of the group generated by inversion in the origin: the group of all translations and inversion in all points; this is the generalized dihedral group of R3, Dih(R3).

Overview of isometries in up to three dimensions

E(1), E(2), and E(3) can be categorized as follows, with degrees of freedom:

Isometries of E(1)
Type of isometry Degrees of freedom Preserves orientation?
Identity 0 yes
Translation 1 yes
Reflection in a point 1 no
Isometries of E(2)
Type of isometry Degrees of freedom Preserves orientation?
Identity 0 yes
Translation 2 yes
Rotation about a point 3 yes
Reflection in a line 2 no
Glide reflection 2 no

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Isometries of E(3)
Type of isometry Degrees of freedom Preserves orientation?
Identity 0 yes
Translation 3 yes
Rotation about an axis 5 yes
Screw displacement 6 yes
Reflection in a plane 3 no
Glide plane operation 5 no
Improper rotation 6 no
Inversion in a point 3 no

Chasles' theorem asserts that any element of E+(3) is a screw displacement.

Commuting isometries

For some isometry pairs composition does not depend on order:

• two translations
• two rotations or screws about the same axis
• reflection with respect to a plane, and a translation in that plane, a rotation about an axis perpendicular to the plane, or a reflection with respect to a perpendicular plane
• glide reflection with respect to a plane, and a translation in that plane
• inversion in a point and any isometry keeping the point fixed
• rotation by 180° about an axis and reflection in a plane through that axis
• rotation by 180° about an axis and rotation by 180° about a perpendicular axis (results in rotation by 180° about the axis perpendicular to both)
• two rotoreflections about the same axis, with respect to the same plane
• two glide reflections with respect to the same plane

Conjugacy classes

The translations by a given distance in any direction form a conjugacy class; the translation group is the union of those for all distances.

In 1D, all reflections are in the same class.

In 2D, rotations by the same angle in either direction are in the same class. Glide reflections with translation by the same distance are in the same class.

In 3D:

• Inversions with respect to all points are in the same class.
• Rotations by the same angle are in the same class.
• Rotations about an axis combined with translation along that axis are in the same class if the angle is the same and the translation distance is the same.
• Reflections in a plane are in the same class
• Reflections in a plane combined with translation in that plane by the same distance are in the same class.
• Rotations about an axis by the same angle not equal to 180°, combined with reflection in a plane perpendicular to that axis, are in the same class.

References

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• William Thurston. Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN 0-691-08304-5